 Excursion weights of a reflected birth-death chain

For a nearest-neighbour chain on the nonnegative integers, reflect at zero with holding probability $q_0$ and upward probability $p_0$. In a recurrent excursion from zero until its first return, expected visits $w_i$ satisfy $w_0=1$ and $w_i{p}_i=w_{i+1}{q}_{i+1}$, because each upward crossing is paired with a downward crossing. Thus the expected excursion length is ${\sum }_i{w}_i$ by Tonelli's theorem. Positive recurrence is equivalent to finiteness of that sum. The hitting-probability scale weights are ${\rho }_n={\prod }_{r=1}^n{q}_r/p_r$, and $w_n{\rho }_n=p_0/p_n$ helps show that summable excursion weights imply recurrence.